🧄Using Stokes' Theorem with a Constant Scalar Field

The problem illustrates the significance of Generalized Stokes' Theorem in relating surface integrals to line integrals, particularly emphasizing that the scalar field $\phi(x)$ must be constant on the boundary curve $C$. This constancy allows for simplification of the line integral, transforming it into a fundamental case where the Fundamental Theorem of Line Integrals applies, ensuring that the integral evaluates to zero. The visual demonstration reinforces that without the boundary condition of a constant scalar field, the integral can yield non-zero results, underscoring the critical role of this condition in achieving a predictable outcome.

🎬A constant scalar field leads to a zero integral result

The demo visually confirms that the condition of a constant scalar field on the boundary is essential for the surface integral to be zero. When the scalar field $\phi$ is constant on the boundary, the line integral evaluates to zero. When $\phi$ is not constant, the line integral has a non-zero value, and the proof fails. This highlights the importance of the initial condition in the problem statement, which turns a potentially complex integral into a straightforward case with a predictable result.

🖊️Mathematical Proof

Using Stokes' Theorem with a Constant Scalar Field | Proof and Derivationviadean on Notion